Quadratic Equations
A polynomial of degree 2, is called a quadratic expression. A quadratic expression has the general form of ax2+bx+c, where a'', ''b and c'' are constants and the variable ''x has the highest exponent of 2'''. For example, ''3x2-5x+7'' is a quadratic expression. Quadratic Equations A '''quadratic equation, or a polynomial equation of degree 2, is an equation of degree 2 where its general form, ''Formation of a Quadratic Expression or Equation'' To form a quadratic expression, we make use of the algebraic identity: *''(x-p)(x-q) = x2+(p+q)x+pq, where '''p' and q''' are roots of the expression. So, if given '''p and q''' as the roots, we can form a quadratic expression by multiplying, *(x-p)(x-q)' ''Solving Quadratic Equations To solve a quadratic equation, means to find the roots p''' and '''q(or the value of x') of the equation. A typical quadratic equation can be solved by using 3 methods: ''Factorisation, Completing the Square or by using the Quadratic Formula. Factorisation To factorise, we find the factors of the quadratic equation. Consider the algebraic identity: *(x-p)(x-q) = x2+(p+q)x+pq' Comparing with the general form 'ax2+bx+c, we need two integers, '''p and q''', both satisfies the following equations: #p+q = b' #pq = c'' Once we found the factors, we write (x-p)(x-q) = 0. By the fundamental theorem of polynomials, we get two solutions x=p or x=q. If we are solving an equation involving the constant a'', we then consider the value of ''a in the equation: #''(5x-p)(x-q) = 5x2+(p+q)x+pq'' However, most of the quadratic equations cannot be factorised, which means that they cannot be solved by factorisation. Now we may use other methods like completing the square or quadratic formula to find the value of x''. Quadratic Formula / Completing the Square The '''quadratic formula' can be used to solve all types of quadratic equations. It can be derived by completing the square, so both are generally the same method. Completing the Square makes use of the algebraic identity: To derive the quadratic formula by completing the square in the equation ax2+bx+c=0: *Make the value of a=1, by dividing both sides of the equation by a''. *Rearrange the equation so that all the ''x terms are on the left side, with the right side for the constant. *Add (b/2)2 to both sides of the equation to complete the square. *The equation then becomes *Write the right hand side as a single fraction. *Taking square roots, we get *Since the objective was to find x, we make x the subject. *The resulting equation is the quadratic formula. Taking the quadratic formula as the root p and q of a quadratic equation, we get the quadratic factorisation formula: Discriminant The discriminant, b2-4ac, which is the expression square rooted of the quadratic formula, determines the type of roots p''' and '''q of the quadratic equation. The following shows the types of roots of a general quadratic equation. *If b2-4ac > 0, the equation have 2 real and distinct roots. *If b2-4ac = 0, the equation have the same real roots. *If b2-4ac < 0, the equation have complex roots, i. For example, we determine the type of roots of the quadratic equation.